For questions about or involving covering spaces in algebraic topology.

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10
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183 views

Quotient Groups and Covering Spaces in Painting Hanging

Consider the $1$-out-of-$n$ painting hanging problem: Given $n$ nails in a wall, how can we hang a painting such that upon removal of any nail, it falls. This has a nice interpretation as a problem in ...
6
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136 views

Are there infinitely many non-negative integers not covered by one of these 7 polynomials?

Consider the following polynomials: $$ \begin{align} f_1(n, m) &= 30nm + 23n + 7m + 5 \\ f_2(n, m) &= 30nm + 17n + 13m + 7 \\ f_3(n, m) &= 30nm + 23n + 11m + 8 \\ f_4(n, m) &= ...
5
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57 views

Covering space of surface of infinite genus

Let $X$ be a surface of infinite genus that is not compact (with edges extending to infinity). How would I show that this is a covering space of the 2-torus $T^{1}\# T^{1}$ via the action of the free ...
5
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69 views

Show that a map of sets is continuous if its composition with other functions is

Problem: Let $Y, E, B$ be topological spaces with $Y$ locally path connected. Suppose $p: E \rightarrow B$ is a covering map, with $g: Y \rightarrow E$ a map of sets. If $p \circ g$ is continuous, ...
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71 views

Covering spaces of Lie groups

In a paper of Tom Bridgeland's, he describes an action by the universal over $G:=\tilde{GL^+}(2,\mathbb{R})$ using a description of $G$ I find unintuitive. Namely, he indexes write the fiber over $T\...
5
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202 views

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper (...
5
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84 views

Can we make topological covers of $\mathbf{P}^1$ minus three points into schemes

Let $k=\overline{\mathbf{Q}}$. Fix a finite closed subset $B\subset \mathbf{P}^1_k$. Let $X$ be a "nice" topological space and suppose that there is a continuous morphism $f:X\to \mathbf{P}^1_k-B$. ...
5
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48 views

How to construct certain cover given in Mumford's Abelian Varities book

In chapter I, appendix to section 2 of the book "abelian varieties" by Mumford, we consider a discrete group $G$ acting freely and discontinuously on a good topological space $X$ (i.e., $\forall x \in ...
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122 views

An entire function with finite covering group is a polynomial.

Let $f$ be an entire function. Think of it as a covering space of $\mathbb{C}$ (perhaps with isolated punctures) to $\mathbb{C}$ (perhaps with isolated punctures). Suppose we know there is only a ...
3
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46 views

Do all paths have a neighborhood about them which lift homeomorphically to a covering space?

Let $\widetilde{X}$ be a covering space of $X$ with projection map $p:\widetilde{X}\to X$. We know by definition that, for all $x$ in $X$, there exists a neighborhood $x\in U\subset X$ such that $p^{-...
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57 views

Euler characteristic of 2-sheeted covering space

I'm currently taking a course on algebraic topology and while doing exercises, I realised that I wanted to use the following: If $X$ is a compact connected $2$-manifold and $\varpi:Y \rightarrow X ...
3
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107 views

covering spaces

A covering space of a Hausdorff space is also Hausdorff. Conversely, a compact Hausdorff finite covering space has a Hausdorff base space. However, in general, a non Hausdorff space may have a ...
3
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162 views

Does the pullback of a covering space correspond to the pullback of the corresponding representation of $\pi_1$?

In other words, suppose you have a degree $n$ covering space $C\rightarrow X$ corresponding to some (equivalence class of) representation $\pi_1(X)\rightarrow S_n$. Suppose you have any continuous map ...
3
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81 views

does the pullback of a covering space correspond to the pullback of the corresponding representations of $\pi_1$?

Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back ...
3
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139 views

Connected Components of Covering Space

Let $\pi : \tilde M\to M$ be a covering map, $K_1, K_2\subset\tilde M$ are two different connected components and there exists such points $x\in K_1, y\in K_2$ that $\pi(x)=\pi(y)$. In other words, $\...
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648 views

Algorithms for covering a rectilinear polygon using rectangles of the same size

The following is the problem description: All angles of the polygon are right. It may be convex or concave. Use rectangles of the same size to cover the polygon. The edge of the polygon and ...
3
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81 views

How do we check if a covering of an orbifold is a manifold?

Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some ...
3
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108 views

Characterization of maps with $\mathbb{Z}/2\mathbb{Z}$-equivariant lifts.

I'm interested in characterizing maps $f:\mathbb{R}P^k\to\mathbb{R}P^\infty$ that lift to a $\mathbb{Z}/2\mathbb{Z}$-equivariant map $\tilde{f}:S^k\to S^\infty$. For $k\geq 2$ I have been able to ...
3
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38 views

Are there generalizations of Prym varieties to higher dimensions

Prym varieties are abelian varieties that are associated to a double cover of algebraic curves. Can we also associate an abelian variety to a double cover of algebraic surfaces in a reasonable way? ...
2
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27 views

If $p: E \rightarrow X$ is a covering map with $E$ connected and $|p^{-1}(x_{0})|=k$ for some $x_{o}$ then $|p^{-1}(x)|=k$ for all $x \in E$.

Prove that if $p:E \rightarrow X$ is a covering map with $E$ connected and $p^{-1}(x_{0})$ has $k$ elements for some $x_{0} \in X$, then $p^{-1}(x)$ has $k$ elements for every $x \in X$. Is my proof ...
2
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54 views

Is this what universal covering spaces are used for?

From the perspective of real analysis, we have: $$\int_{-1}^{1}\frac{1}{1+x^2} = \mathrm{tan}^{-1}(1)-\mathrm{tan}^{-1}(-1) = 2\mathrm{tan}^{-1}(1) = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2}$$ Something ...
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65 views

limit of covering spaces

Say we have $X$ a manifold with a compact exhaustion of embedded submanifolds $X=\cup K_n$ with $K_n\subset K_{n+1}$. Let $H\subset \pi_1(X)$ a infinite index subgroup that is finitely generated, ...
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20 views

Maximal analytic continuation gives rise to a covering

Suppose that $a$ is a point on a connected Riemann surface $X$ and $\varphi \in \mathcal{O}_a$ admits an analytic continuation along every curve in $X$ starting at $a$. Let $(Y, p, f, b)$ be the ...
2
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72 views

3 sheeted cover of Klein bottle with torus

So I'm dealing with this exercise in which it is asked to determine whether the torus can be a 3-cover of the Klein bottle. A friend of mine came up with a proof that this is not the case, but this ...
2
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48 views

How to show explicitely that 2-sheeted covers are Galois?

Let $X,Y$ be connected Hausdorff topological spaces. It is well-known that every 2-sheeted covering $p:Y\to X$ is Galois which means that $Aut(Y/X)$ acts transitively on fibers. It is easy to come up ...
2
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209 views

Show that if $f$ is a proper surjective map which is locally injective then $f$ must be a covering map

Suppose $f :X \to Y$ is a continuous proper map between locally compact Hausdorff spaces. Show that if $f$ is a surjective map which is locally injective then $f$ must be a covering map. It is well ...
2
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42 views

About prime geodesic cycles and deck transformations group

I'm proving theorem 2 occurring in Sunada's paper Riemannian coverings and isospectral manifolds. Unfortunately Sunada's quotes himself to the following paper: Tchbotarev’s density theorem for closed ...
2
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0answers
214 views

universal cover homotopy equivalent if the base space homotopy equivalent

I am working on Hatcher's algebraic topology book and I got stuck in problem 8 in section 1.3. It says if $\hat{X}$ and $\hat{Y}$ are simply-connected covering space of the path connected, locally ...
2
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85 views

Uniqueness for a covering map lift: is locally connected necessary?

So I just got through proving the following theorem: If $p:C\to X$ is a covering map and $Y$ is a [xxx] space, then given $y_0\in Y$, $c_0\in C$, $f:Y\to X$ such that $f(y_0)=p(c_0)$ there exists ...
2
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0answers
48 views

Morphism induced in cohomology of a covering space

It is a basic question but I'm stuck. If $p:M\rightarrow N$ is a $m$-fold unramified covering between surfaces, why the morphism induced by $p$ in cohomology at level 2 with coefficients in $\mathbb{...
2
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0answers
125 views

Unique Path lifting of covering map

Let $p:E\rightarrow B$ be a covering map (in particular $p$ is a fiber bundle with discrete fiber). We want to prove the following: Given a commuting diagram of the following form: $\{0\}\rightarrow ...
2
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0answers
46 views

A space having exactly three coverings up to equivalence

Q: Give an example of a topological space having exactly 3 coverings up to equivalence (including a covering by the space itself). Proof: There is a theorem that says that given a topological ...
2
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0answers
110 views

Branched coverings of unit disk

Is there a classification of branched coverings of the closed unit disk $\mathbb{D} =\{z\in \mathbb{C} \ | \ |z| \leq 1 \}$? Here we consider only branched covering projections which restrict to ...
2
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78 views

The cohomology of $S^3/D^*_k$

I have tried to compute the de Rham cohomology and the homology over the integers of the space $S^3/D^*_k$, where $D^*_k$ is the binary dihedral group of order $4k$ and I would like to know if what I ...
2
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0answers
98 views

A sufficient condition for the composition of covering maps to be a covering map

Let $q:X \rightarrow Y$ and $r:Y \rightarrow Z$ be covering maps and $p= r \circ q$. If $r^{-1}(z)$ is finite for all $z \in Z$, then $p$ is a covering map. Now I found the following proof: Let ...
2
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0answers
187 views

Classifying covering spaces of product spaces

Given two covering maps $p\colon \tilde{X} \to X$ and $q\colon \tilde{Y} \to Y$, we can form the covering map $p\times q \colon \tilde{X} \times \tilde{Y} \to X\times Y$. By covering space theory, we ...
2
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0answers
65 views

lifting injective maps to injective maps in principal bundles

Let $i \, :Y \hookrightarrow X$ be an inclusion of (nice) topological spaces, and suppose that the induced map $\pi_1(Y) \to \pi_1(X)$ is injective. Then every lifting of $i$: $$ \tilde Y \to \tilde ...
2
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332 views

what is the covering space of figure eight which is corresponding to commutator subgroup.

Let $ F$ be the free group on two generators and let $F^{'}$ be its commutator subgroup. Find a set of free generators for $F^{'}$ by considering the covering space of the graph $S^{1} \vee S^{1}$ ...
2
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0answers
37 views

Correct definition of a regular covering without global connectedness hypotheses

Let $p:Y\to X$ be a covering map of topological spaces where $X$ is assumed to be locally path connected (and hence the same is true of $Y$) but neither $X$ nor $Y$ is assumed to be connected. In this ...
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0answers
31 views

Submersions and complex structure

Let $f : \Lambda \rightarrow X$ be a continuous surjective map, where $\Lambda$ is a complex manifold and $X$ a topological space. Suppose that for all $x \in X$, there is a neighborhood $U_x$ of $X$ ...
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77 views

Covering Spaces and Fundamental Groups

Can somebody tell me if what I did is right? I need to Draw the based cover $\hat{B}\rightarrow B$ such that $\pi_{1}(\hat{B},v)$ corresponds to the subgroup $\langle a^{3}, a^{2}b\rangle$ $\...
2
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0answers
52 views

Degree and picture of a Covering map

I need to know if I am right: I need to know the degree of this covering map $R \rightarrow S$: $T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\#T^{2}\ \rightarrow T^{2}\#T^{2}$ I have that genus of $R$, $g_{R} =...
2
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0answers
45 views

Reference request for an explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface

Let $\Sigma_g$ be a geuns $g$ Riemann surface with $g \geq 2$. It can be thought of in the following way: it is the quotient space $$\mathbb{H}/\pi_1(\Sigma_g)$$ where an element of $\pi_1(\...
2
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0answers
59 views

Are there Galois covers of curves branched at 1 point?

Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one ...
2
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0answers
112 views

Why is the pullback of a connected cover not necessarily connected?

In particular, I read somewhere that the fiber product of the maps $S^1\rightarrow S^1$ sending $z\mapsto z^m$ and $S^1\rightarrow S^1$ sending $z\mapsto z^n$ is disconnected with $\gcd(n,m)$ ...
2
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0answers
331 views

Restriction of a covering map to a subspace

Let $p:X\rightarrow Y$ be a covering map and let $Y_0 \subset Y$. Show that $p|:p^{-1}(Y_0)\rightarrow Y_0$ is a covering map. Hint: Show first that if $V\subset Y$ is well-covered by $p$, then $Y_0\...
2
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0answers
207 views

Example of a nontrivial finite covering map

A covering map $p:C\to X$ is called finite when for each $x\in X$ the fiber of $x$ is finite. I have to prove something about such covering maps, but I have never seen a nontrivial example of one. ...
2
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0answers
70 views

How to find every 4-sheet covering of the wedge sum?

Based on this question 4-sheet covering of the wedge sum of two circles I know how to find one 4-sheet covering of the wedge sum, but how to find every 4-sheet covering of the wedge sum? I really ...
2
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0answers
88 views

Global sections of covering spaces

Let $p:C\to X$ be a covering space having a global section $s:X\to C$. I can show that this implies that $s(X)$ is disconnected from the rest of $C$. Is there any reference where this is explicitly ...
2
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0answers
123 views

Are there infinitely many rational functions of bounded degree and given ramification

It is well known that the set of branched covers $X\to \mathbf{P}^1(\mathbf{C})$ of bounded degree and given branch locus is finite (up to isomorphism). Edit. The branch locus $B$ of $f:X\to \mathbf{...